Signal Processing and Linear Systems-B.P.Lathi copy

23a 223b t he initial conditions are see eq 221 for

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Unformatted text preview: his section is devoted to the determination of t he zero-state response of a n LTIC system. T his is t he system response y(t) t o a n i nput f (t) when t he s ystem is in zero state; t hat is, when all initial conditions are zero. W e s hall a ssume t hat t he s ystems d iscussed i n t his s ection a re i n z ero s tate u nless m entioned o therwise. U nder these conditions, t he zero-state response will be t he t otal response of t he system. We shall use t he s uperposition principle here t o derive a linear system's response t o some a rbitrary i nput f (t). I n this approach, we express f (t) in terms of impulses. We begin by approximating f (t) w ith narrow rectangular pulses, as depicted in Fig. 2.3a. This procedure gives us a staircase approximation of f (t) t hat improves as pulse w idth is reduced. In t he limit as pulse width approaches zero, this representation becomes exact, a nd t he r ectangular pulses become impulses delayed by various amounts. T he s ystem response t o t he i nput f (t) is t hen given by t he s um of t he s ystem's responses t o each (delayed) impulse component of f (t). I n o ther words, we c an d etermine y(t), t he system response t o any arbitrary i nput f (t), if we know t he i mpulse response of t he system. For t he sake o f generality, we place no restriction on f (t) as t o where it s tarts a nd where it ends. I t is therefore assumed t o exist for all time, s tarting a t t = - 00. T he s ystem's t otal response t o t his input will t hen be given by t he s um of its responses t o each o f t hese impulse components. This process is i llustrated in Fig. 2.3. Figure 2.3a shows f (t) as a sum of rectangular pulses, each of width f:::.r. In the limit as f:::.r --+ 0, each pulse approaches a n impulse having a strength equal t o the a rea u nder t hat pulse. For example, as f:::.r --+ 0, t he s haded rectangular pulse located a t t = nf:::.r in Fig. 2.3a will approach a n impulse a t t he same location with , trength f(nf:::.r )f:::.r ( the shaded area under t he r ectangular pulse). This impulse ~an t herefore be represented by [J(nf:::.r)f:::.r]8(t - nf:::.r), as shown in Fig. 2.3d. I f t he s ystem's response t o a u nit impulse 8(t) is h(t) (Fig. 2.3b), its response to a delayed impulse 8(t - nf:::.r) will be h(t - nf:::.r) (Fig. 2.3c). Consequently, ; he s ystem's response t o [J(nf:::.r ) f:::.r]8 (t - nf:::.r) will be [J(nf:::.r )f:::.r]h(t - nf:::.r), as 1lustrated in Fig. 2.3d. These results can b e conveniently displayed as i nput-output Jairs with a n a rrow d irected from t he i nput t o t he o utput as shown below. T he leftland side represents t he i nput, and t he r ight-hand side represents t he corresponding ;ystem response: 8(t) = =} h(t) (a) I 1-- I=nll,; h ( I) II ( I) o 1-- (b) II ( t - nil,;) o n il,; h ( I - nil,;) o 1-- n il,; t-- (c) • [ f(nll,;) 1 l"t]1I ( I - nil,;) o n il,; [ (nil,;) h ( I - nll,;)Il,; l ly(/) o 1-- n il,; 1-- (d) (e) 8(t - nf:::.r) = =} h(t - nf:::.r) Jf(n.6r)f:::.r!8(t - nf:::.r), = =} J f(nM)f:::.rlh(t - nf:::.r), i nput o utput (2.27) F ig. 2 .3 F inding t he s yst...
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